Preliminary Note: On a keyboard instrument like a piano, the thing that you push down with your finger, we call a digital. Each digital is connected to a series of action parts which transfer your finger motion to a swinging "hammer" which strikes either one string, two strings tuned in unison, or three strings tuned in unison. The string or strings vibrate and produce a tone having a specific pitch. Most pianos have 88 digitals which produce 88 distinct pitches.
Computer: Human, working from pitch number 49, to which you have assigned a value of 440 hz, I have calculated 88 pitches. Of those, pitch number 40 is 261.6255653006 hz.
We can extend our piano keyboard beyond the usual 88 pitches. If you entered a pitch number above 88 or below 1 (you can enter negative numbers), you won't see its frequency listed above, but I can calculate and display its frequency for you here. Pitch number 40 is 261.6255653006 hz.
pitch 1 is 27.5000Computer: I work fast, don't I? Most likely I calculated and displayed all 88 frequencies in less than a second. You know how long this would have taken you, human, if you had had to do it with an electronic calculator and a typewriter or pen? Probably a good few hours. You know how long this would have taken you, if you had had to do it without any kind of electronic or mechanical computing device (not even a slide rule) to aid you? Probably several days of working 8 hours per day.
LeafyGreen, Computer Programmer: The php program I wrote, copyright 2006 by Theodore Zuckerman, that calculates the frequencies, the pitches, is a server-side program, so you won't see it in the source document for this page. The calculations are done on the web server computer, then just the results are sent to your computer. If you are interested in seeing a copy of the source code, please contact me. I should add that to start with one pitch, multiply it by the twelfth root of two, note the result, multiply that by the twelfth root of 2, note the result, multiply that by the twelfth root of 2, etcetera, would result in the accumulation of imprecision. To avoid that, I calculated each pitch using a formula that related it to pitch 49.
Here is a sketch of a piano keyboard with each digital numbered (from 1 to 88) and labeled with the pitch it produces. I made that sketch around 1976, before I owned a personal computer. I calculated the frequencies, one by one, with an electronic calculator. Rather than provide you with a color-corrected image, I have provided an image which allows you to see how the paper that the digitals were drawn on, has yellowed more than the paper, with the frequencies on it, that I pasted it to. I'm not sure whether this is an unnecessarily nasty-looking bit of business, or a charming bit of memorbilia that adds character to my web site.
You might be interested in Leafy Green's Excel spreadsheet which, in addition to all 88 pitches, shows all the beat rates for many equally tempered intervals — fifths, fourths, major thirds, minor thirds, major sixths, minor 6ths, 10ths, etcetera, from one end of the piano to the other. Positive beat rates indicate expanded intervals (the pitches are further apart than if they were justly tuned), negative beatrates, contracted. Now, the on-line version that you open in your browser is a fixed rendition of the original Excel spreadsheet, and its beat rates are based on a fixed value for A49, 440 hz, but if you open the file in Excel you will be able to calculate how all the beatrates (and pitches) would change if the value of A49 were changed. This could be useful if you want to tune an instrument to equal temperament, without first raising its pitch to standard pitch, or to tune a historic intrument that was designed to sound and work best at A436 or whatever, instead of at the contemporary standard of A440.
Page created by Leafy Green Web Publishing
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